Group Structural Realism (Part 3)

Can groups do a better job at surviving theory change than individual objects? Here’s one observation that might make us think so: groups are often insensitive to a change in underlying set. So, it’s possible for the group structure of an early scientific theory to be preserved in a later theory, even if the descriptions of objects are not.

A toy example: Imagine that some theory leads us to propose the existence of a cube. Suppose that later, we discover that there is no cube, but rather an octahedron.This theory was wrong about what kinds of objects exist. However, it was right about the group-structure, since cubes and octahedrons have the same symmetry group (rotations of pi/2 about appropriate axes preserve the orientations of both objects; so do flips about an appropriate plane). So, if we were betting on which item would be preserved under theory change, a bet on groups would have won out over a bet on objects.

Two more relevant examples:(1) Consider the change from the Galilei group to the Poincaré group, ushered in by special relativity. At first blush, this appears to be a discontinuity of group structure over theory change. As we discussed in the previous post, Wigner’s legacy allows for a theory of ‘Galilei particles.’ However, the group of Galilei transformations predicts the wrong kinds of particles (i.e., the wrong momentum eigenvalues), as well as the wrong commutation relations. Consequently, in the transition to the Poincaré group, the taxonomy of fundamental particles changed.

However, ‘Galilean particles’ do happen to have the right angular momentum quantum numbers – they allow for the existence of spin, for example. A realist about particles has little to say about this fact. But GSR can actually provide an explanation: it is because rotation group is what’s metaphysically fundamental about angular momentum, and the rotation group SO(3) was preserved in the transition from the Galilei to the Poincaré group — as a subgroup of each. As for the Galilei group as a whole, one might say that it was also preserved in approximate form, in low-velocity regimes.

(2) This same rotation group SO(3) provides yet another example of preservation under theory change. With the discovery of spin, the traditional realist should seemingly admit that a new kind of particle was discovered, signifying a discontinuity over theory change. But, according to GSR, the important change was really the extension of the symmetry group SO(3) to a larger group, SU(2). The latter is the correct rotation group for a quantum theory of spin, because it admits j = 1/2-integer representations. However, SO(3) is not rejected in this correction – it is preserved as a subgroup of SU(2).

The point of these examples, for the budding structural realist, is to suggest that group structures – not individual objects, and not even algebras of observables – are the superior candidates for the survival of theory change. If this turns out to be right, then GSR not only provides a natural, precise example of structural realism; it also stands a promising chance of satisfying the original, ‘pessimistic meta-induction’ motivation for structural realism.

I hope to have argued (maybe a bit too long-windedly) that there is a fairly precise and compelling form of structural realism in quantum theory: Group Structural Realism. Unfortunately, I don’t think it works. As we’ll see in the next (and final) post, GSR actually illuminates a general objection to structural realism, which I call the `higher structures’ objection.